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Isomorphism factorizations of the complete graph into Cayley graphs on CI-groups

Huye Chen, Jingjian Li, Hao Yu, Zitong Yu

Abstract

Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this paper, we investigate isomorphic factorizations of complete graphs into Cayley graphs on CI-groups. Let $Γ=Cay(G,S)$ denote the Cayley graph of finite group $G$. We obtain a necessary and sufficient condition on CI-group $G$ so that the complete graph on $|G|$ vertices can be edge-partitioned into $k$-copies of Cayley graph of the same CI-group $G$ each isomorphic to $Cay(G,S)$ for some inverse-closed subset $S\subset G\setminus\{1\}$. Further we give a construction of isomorphic factorizations of the complete graph into Cayley graphs on CI-group.

Isomorphism factorizations of the complete graph into Cayley graphs on CI-groups

Abstract

Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this paper, we investigate isomorphic factorizations of complete graphs into Cayley graphs on CI-groups. Let denote the Cayley graph of finite group . We obtain a necessary and sufficient condition on CI-group so that the complete graph on vertices can be edge-partitioned into -copies of Cayley graph of the same CI-group each isomorphic to for some inverse-closed subset . Further we give a construction of isomorphic factorizations of the complete graph into Cayley graphs on CI-group.
Paper Structure (7 sections, 19 theorems, 8 equations, 1 table)

This paper contains 7 sections, 19 theorems, 8 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $k$-rotational group
  4. Proof of Theorem \ref{['mainCI']}
  5. Case 1
  6. Case 2
  7. Case 3

Key Result

Theorem 1.1

Let $G$ be a CI-group. Then $G$ has $k$-if property if and only if $G$ is the direct product of elementary abelian group, and the order of each Sylow subgroup $G_p$ of $G$ satisfy $2k\mid |G_p|-1$ when $p$ is odd; $k\mid |G_p|-1$ when $p=2$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 25 more